A Bayesian deep prior-based quaternion matrix completion for color image inpainting

Protocol for natural and biomedical image processing in the hypercomplex domain using the 2D orthogonal planes split.

<bold></bold> A quaternion n × n matrix Q is Hermitian R -conjugate if R Q R = Q ¯ , Q ∗ = Q for some orthogonal symmetric matrix R ≠ ± I . This paper presents finite iterative algorithms for solving two-sided k -conjugate quaternion matrix equations with Hermitian R -conjugate solution. When the equation is consistent, the convergence theorem guarantees a solution within a finite number of iterations, assuming no round-off errors, for any initial arbitrary Hermitian R -conjugate solution. Finally, two numerical instances demonstrate the theoretical results and impact of the proposed algorithms.

This paper extends the theory of equivalence canonical forms from quaternion matrices to quaternion tensors under the Einstein product. Motivated by recent results on the simultaneous decomposition of two specific configurations of five quaternion matrices, we establish a comprehensive framework for the corresponding configurations of five quaternion tensors. The core approach leverages bijective transformation maps that establish isomorphisms between quaternion tensor spaces and matrix spaces, allowing us to systematically construct invertible transformation tensors that simultaneously reduce the given tensor quintuples to canonical forms consisting solely of binary entries (0 and 1). A detailed structural analysis of the resulting canonical tensor forms is provided, including explicit dimension formulas for all identity blocks derived from precise rank conditions. To demonstrate practical utility, we integrate the proposed tensor decomposition with the discrete wavelet transform to construct a color video encryption and decryption system. Experimental results confirm perfect reconstruction (PSNR exceeding 300 dB, SSIM equal to 1) and strong security performance: NPCR of 49.8%, UACI of 49.6%, information entropy of 0.9986 bits per pixel, adjacent pixel correlation below 0.03 in absolute value, and a key space exceeding 2512. The developed theory significantly extends the existing literature on quaternion tensor decompositions and provides powerful tools for multidimensional signal processing.

In this paper, we study the Cholesky and Cholesky-QR decompositions of dual quaternion matrices. Firstly, the dual quaternion Gauss transformation matrix is given as the basic tool to realize these two decompositions. Secondly, it is proved that a dual quaternion Hermitian matrix with positive definite standard part can be performed the Cholesky decomposition, and the corresponding direct algorithm is given. Moreover, if the standard part of a dual quaternion matrix is column full rank, the Cholesky-QR decomposition can be realized. Inspired by the real structure-preserving algorithm of quaternion matrices, we design a real calculation-preserving algorithm to realize the Cholesky decomposition of dual quaternion matrices. Numerical experiments show that the real calculation-preserving algorithm is more effective than the direct algorithm. Finally, a concrete example of applying the Cholesky decomposition of dual quaternion matrices to the Cholesky-QR decomposition of dual quaternion matrices is given.

Quaternion version of biconjugate residual method for generalized coupled Sylvester-type quaternion matrix equations with application to color video encryption and decryption

Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications

Pass-efficient randomized algorithms for low-rank approximation of quaternion matrices

Nonlinear hierarchical quaternion matrix factorization-based quaternion matrix completion

Real projection algorithms for generalized low-rank approximation of large-scale quaternion matrix in color image processing

Solving a system of quaternion matrix equations by using PSVD for multiple matrices with applications

Reweighted low-rank quaternion matrix factorization with deep denoising prior for color image inpainting

This paper introduces a finite iterative algorithm for solving a single-variable quaternion system of matrix equations. It is theoretically proven that if the system is consistent, the solution can be achieved from any initial quaternion matrix within a finite number of iterations, assuming the absence of round-off errors. A special case of this equation is studied, which contains the k -conjugate of the unknown matrix. Numerical examples are provided to illustrate the effectiveness and computational efficiency of the proposed algorithms.

Nested Quaternion Matrix and Recursive Approach to Its Singular Value Decomposition

Quaternion matrix completion with total variation regularization utilizing fast dual proximal gradient method

ABSTRACT Robust quaternion matrix completion (RQMC), which aims to recover clean data from data that is both incomplete and corrupted by noise, has recently attracted extensive attention in the fields of image and signal processing. This problem can be iteratively solved by applying the close‐form solution of proximal operator, including quaternion singular value thresholding (QSVT) operator. However, the computational complexity of the QSVT operator is extremely high, which seriously affects the solution efficiency of the RQMC model. In this paper, we propose a randomized algorithm for the RQMC model based on the randomized low‐rank approximation technique. Then, through theoretical analysis, we prove that the randomized algorithm offers an ideal approximation of the deterministic algorithm. Finally, through some numerical experiments, we demonstrate the effectiveness and reliability of the proposed algorithm.

ABSTRACT In high‐dimensional data processing, quaternion generalized singular value decomposition (QGSVD) has become one of the most important solvers for advanced mathematical models. However, there are currently fewer efficient algorithms for calculating partial quaternion generalized singular values and vectors. When dealing with a quaternion matrix pair of substantial size, we present a stable QGSVD and two randomized algorithms cooperating with structure‐preserving and quaternion sampling strategies, and generate good low‐rank approximations to the original quaternion matrix pair. Explicitly, by leveraging random sampling based on the quaternion normal distribution for the original quaternion matrix pair, we propose both fixed‐rank and adaptive randomized algorithms for QGSVD. We also establish different kinds of error bounds for the randomized QGSVD approximation in the theoretical analysis and illustrate the effectiveness of the randomized QGSVD algorithms by numerical experiments on simulation data and practical color face recognition.

In the era of big data and multimedia communication, securing color images against unauthorized access and attacks is a pressing challenge. While quaternion-based models provide a unified representation for color images, most existing encryption schemes rely on single-image frameworks or lack the mathematical rigor to ensure both security and feasibility. To bridge this gap, this paper introduces a system of generalized Sylvester-type quaternion matrix equations as a novel encryption model. By using the equivalence canonical forms of five matrices arranged in a specific array, we provide necessary and sufficient conditions for the solvability of the generalized Sylvester-type quaternion matrix equation system, depending on the rank of the coefficient matrix. Numerical examples are provided to validate the obtained results. As an example of applications, we develop an encryption scheme for color images based on the proposed quaternion matrix equation system. Experimental results confirm the high feasibility of the proposed scheme. Notably, the proposed model supports dynamic key updates and multi-image secure transmission, making it highly adaptable for real-world applications. By integrating advanced quaternion matrix theory with practical image encryption, this work offers a scalable, secure, and mathematically sound approach to color image protection.

A nonlinear conjugate gradient method for solving quaternion matrix equation and its application in color image encryption and decryption

Constrained low-rank approximation of quaternion matrices and beyond